My answer to your question is, in short, yes. Of course.
Look at work by Seamus Heaney, Robert Pinsky, etc. Look at the pages of the New Yorker, or poetry quarterlies and you’ll see rhyme and rhythm are certainly alive and well.
In fact, with few exceptions (concrete poetry being one) rhythm and sound are the most important technical considerations in all poetry, free or structured. After all (again with very few exceptions), poetry is meant to be read aloud. It is an art that is based partly on the sound of words.
But be very careful in how you handle rhyme and rhythm. If your poems are very sing-songy and very heavy on the rhymes, you might be introducing an unwanted element of levity into the poem. Or if you never vary your meter the slightest bit, you may be lulling your reader into sleep. For a good example of effective use of unwavering iambic tetrameter, read “Stopping By Woods on a Snowy Evening” by Robert Frost. IIRC, Frost never changes the beat, but there is a reason to it. The sound of the poem reflects the serenity of the scene he is describing. Otherwise, pretty much every single poem you read in classical verse varies its feet from time to time. An poem in iambic pentameter is never iambic all the way through. Spondees or trochees are thrown in to stress words or create pauses. Anapests are thrown in to speed up a line; an extra 11th unstressed syllable is added, etc…
If you know this already, and feel comfortable with this, you’re well on your way.
If there is a trend in rhyme and structure in classical poetry forms today, it’s that of using near-rhymes and judiciously enjambing lines. Seamus Heaney is a prime example of this. Among his works, you’ll find sonnet sequences, terza rima, epic verse and the such with a slightly more modern interpretation of rhyme, in which words like “black/block” or “lake/bleak” are interspersed with more conventional pairs like “clear/year.”
I think the idea that modern poetry is all about esoterism and cleverness removed from emotion is a pile of bullshit. Read some Seamus Heaney, some Elizabeth Bishop or some translated works of Wislawa Szymborska, and then tell me this is the case. I think many people have misconceptions of modern art of all mediums, beit visual, musical or literary, which is based on popular perception rather than by understanding or familiarity with the works themselves. And it’s disheartening for me to see.
At the same time, there are those who eschew convention without understand how the conventions worked and why they’re dismissing them. This is perhaps an even greated mistake.
Otherwise, I think you’ve gotten some great advice here. Good poetry is still good poetry. You just have all the more tools at your disposal, which makes it even more confusing, yet liberating.