What you're asking for can merely solved by shifting the reference frame.

• "I hear you've been under a lot of stress lately"

...is a way of phrasing that shows bad understanding of the medium. "I hear" will most likely irk a lot of people by itself since it's a cheap shot at "See how good I know you?" without being believable. And, as you understood correctly, mentioning stress when trying to get someone to relax will accomplish the opposite in a lot of cases.

• "I hope you don't have too many worries."

This one is a tick better, but not by much. Using "I hope" suggests that the person using it doesn't know whether what they're doing is helping us relax. It conveys a weak frame of mind - which isn't what we want to suggest. And then, well, worries are mentioned again.

I'd like to take two more examples posted below which are about different things than you're talking about.

• /u/AstralD: I suffer from depression and anxiety attacks and being told calmly to let the stress go really helps.

If we're looking for a great way of phrasing things we can achieve these ideas while keeping it positive.

• Bad example to illustrate the point: "Now is the time to calmly let go of all those horrifying disgusting things that stress you out on a daily basis." - nopenopenope.

• Good example: "And now, slowly, you can feel this calmness spreading while you listen to my voice. And while you keep listening this comforting, serene, feeling will make you more and more relaxed and genuinely... happy. You might even start smiling a little bit."

Same idea. I took a more trance-y approach because it feels appropriate but the point I'm trying to make is that instead of "let go of stress" I can suggest to "relax". I can convey "calmness" by using more words that we associate with the same general concepts. Slowly, comforting, calm, serene, relaxation, happiness.