As a jazz-head piano/electronic composer, I tend to end up stacking huge amounts of extensions on my chords. Whenever I use standard non-extended triads they sound sorta empty and boring, but sometimes I end up using so many extra notes that my chords become more a wall of sound than a harmonic progression.
What I'm asking is, how do I get more "oomph" out of standard triads and make them feel "fuller"?
Well, if we can't add extra notes, we still have some options.
The biggest potential change is the voicing of the chords. Some will sound small and thin, and others will sound large and resonant. To my ears, big open fifths can fall into the latter category. I'd recommend sitting down in front of a piano/keyboard, and experimenting. Don't add any extensions! Another option is to study sheet music for more voicing ideas. Another word to look out for here is inversion. Different inversions are basically different voicings of the chord.
On a related subject, the range in which you play the chord will have a big impact. Low octaves are great for imparting power, but they're easy to overuse. Experiment here as well.
It's also worth saying that sometimes less is more. I know it's a cliche, but there's a reason for that. If you're always playing loads of notes, you'll tire your listeners, and they will lose their impact. You do not have to play lots of notes and complicated harmonies to make good music. For me, it takes conscious effort to play less, but I'm almost always happier with the results.
If you feel like cheating a little, seconds (or ninths) are wonderful things. You can sneak them into all sorts of basic triads without anybody getting too upset. Minor sevenths can also be slipped in here and there. Sure, they are extensions, but they're certainly not jazzy.
Moving further into this territory, you could also experiment with different bass notes. A common technique to build tension is to use a constant bass note under moving chords. For example, there's a song I play that has the following bridge progression: G F G Dm7 G F G F. Nothing special. On the second repeat, I play G F/G G Dm7/G G F/G G F. The resolution of tension created by the final F and subsequent chorus chord (C, the tonic) makes this second repeat sound much bigger.
I would recommend looking into guitar approaches a bit, as well as some voice leading.
When guitar players play triads, they are usually hitting the same note a few times, such as an E major in open position: EBEG#BE. Three Es in there and two Bs. Focusing on doubling the root and fifth of the chord brings a lot of depth and support to the "smaller" sound that a triad has compared to one with Jazzy extensions and alterations. Lots of Rock music uses Power Chords (Root, 5, octave) and a lot of distortion, which makes the chords very thick. This of course leaves out the major/minor aspect but it is typically easily determined in the context of all of the chords taken together, included in the melody, or just plain not necessary for the style.
Voice leading can also be very important, depending on what you're going for. The idea of voice leading comes from the concept of Counterpoint, which is the idea of having different Lines or Voices all playing their own part, with the goal of maintaining independence for each part. Basic voice leading includes not using parallel fifths or octaves from chord to chord, eg, if you're going from G to D, you wouldn't want to have two voices playing the note G and both move up to the note A. Parallel fifths and octaves basically destroy the independence of the voices, so if you have a four part harmony and have parallel fifths between two chords, it will sound like you've lost a voice when you arrive at the second chord. I'm not 100% sure but I believe this has to do with overtones. Since fifths and octaves are the first two overtones, they are very closely related, so it's easy for the higher voice to "disappear" into the lower voice because it blends in and sounds like it's just an overtone. Using Contrapuntal voice leading can allow you to have fuller sounding harmonies by maintaining this voice independence. It's important to note the difference between parallel octaves and doubling. If you listen to orchestral works, you will very often hear instruments doubling each other in octaves, which would not be breaking these "rules". The idea is that those two instruments are actually playing the same voice and just adding texture and depth to the harmony, so they're not actually supposed to have the independence that the "rules" are meant to maintain.
These voice leading approaches often lead to solutions that involve inversions and, as others have mentioned, inversions can bring a lot of texture to your chords, sometimes feeling like they've completely changed to another chord. In my experience as a bass player, it can be touchy to use inversions sometimes and it's easy to over use them. It took me a little while to "mature" and understand that there are times that inversions are just not appropriate for the style of music I'm playing, whatever that style may be at the time.
You can also look at some guitar stuff to see patterns of voice leading embedded. You would be looking for songs that have mostly open chord voicings. A lot of progressions that use the standard open voicings follow pretty good voice leading rules and lead to fuller sounding harmonies.
As others have mentioned, exploring range is also valuable. Having strong bass notes adds a lot of depth to your harmony. It's also worth mentioning that the further apart your voices are, the less they sound as though they are working together. The Classical approach within a four part harmony is to have no more than two octaves between your bass and soprano voices (lowest and highest voices). This shouldn't be considered a rule by any stretch but working out parts within a couple octaves of each other may help you hear the cohesiveness that it can offer. From there, you can basically move things up and down and further apart as you wish and hear how it impacts the cohesiveness and stability of your harmony. Rock music is often using the bass player to double the root of each chord an octave below the guitar which adds a lot of depth, so that can help a lot as well.
There's also a lot to be said for context and texture change. We often hear pieces that include a smaller sounding arrangement for an introduction, then the next section will include a larger arrangement, such as guitar only intro/verse with band entering at the first chorus. This can make the larger arrangement sound bigger than it actually is because of where it is coming from. I've used a similar tool in my writing where I will play all of my bass notes an octave up from what you might expect a bass player to be doing, then toward the end of the song at some critical juncture, I'll drop down that octave and everything sounds huge for that important part of the song, furthering the climax.
There are certainly a lot of possibilities as to what may help you achieve your goals here, so it's definitely a good idea to try a lot of them. I also recommend finding some songs to learn that are mostly triadic and try to determine what it is about their approach that works better than what you've been able to do up to now.
It may be obvious to you but it might be worth remembering that triads don't have to be your "basic unit" of arrangement - you can write music that never plays two notes together at the same time and still has a full sense of harmonic progression. Then there's the 'bassline + melody' model, which is full of options, or playing dyads (two notes) instead of triads.
You might think it odd to suggest playing fewer notes if you want a fuller sound, but often more open arrangements give you more room for interesting dissonances and can be very 'nimble' - you can change the whole harmony with just one note. They also mean that when you do throw in a triad, it will sound full in comparison!
When it comes to actual triads, endorph has mentioned them but they're so good I'll say it again : the sus2 chord as a colour in its own right (not as an actual suspension) is very useful and is an important part of pop music vocabulary. It also works well as a base for further extension.
'less is more' can also work when thinking about extensions - you don't have to play all those extra notes at once! When playing in C minor, I might hit a little not cluster of C, D, and E♭ - I'm sort of thinking of it as a Cm9 (no 5 no 7) but playing it as something much simpler. So if you really want to play triads, playing triads that are reductions of extensions can work. You could think of the sus2 chord is an inversion of a 9 (no 3 no 7) chord.
Finally, if you're an electronic composer, you presumably play synths sometimes and you know how a single, steady note on an analogue synth with interesting filter modulation can be a harmonic adventure in its own right; try playing three notes in different octaves with a PWM sound and filter resonance just breaking ito oscillation, then sweep the keytrack - I think your neighbours will agree you can get quite a full sound with just three notes :)
