Hi! I wouldn't, I was a grad assistant and had students ask me to write them letters and told them a letter from a professor would be better, which it is. Just go with the professors, if the professors don't know you well enough, then they could have the grad assistant write the letter and them sign it. Letters from TAs or GAs won't hold any weight.
If you can get the letter co-signed by a prof, that'd be nicer. As a supplement to a professor's letter, I don't think it'll be a problem. Just don't send over 6 letters to a school unless you get permission - even at 6, you risk irritating them by forcing the adcom to read a lot of material.
For what it's worth, I used a letter by a grad student who knew me well, co-signed by a prof. Currently, I've got 2 acceptances, 1 waitlist + defer, 1 top ten waitlist.
I prefer to think of letters as serving a purpose. You don't want to send a school ONLY letters from science professors who can only talk about how good your work was for one class. You want to send a mix. Consider getting letters from people who've watched you grow up from grubby-handed freshman to grizzled senior, maybe one from your job, or perhaps from a humanities class you loved.
Getting into medical school isn't just about showing the admissions committee how awesome your GPA is. While that's incredibly, and sometimes, the most important thing, you stand a better chance if you can also show the committee a face behind the name. You're not Applicant X, 3.62, 29O. You're Firstname Lastname, a guy/girl who loves music/running/panda bears/slam poetry/rainy days/serving up meals at the shelter/short essays/long essays/etc. If you can get letters that can help show off your personality and other intangibles that have no place on a secondary application, you'll be better off... unless your personality is horrible.
I second what StarSup posted. Graduate students dont hold much clout and most dont really have any experience writing letters like this. I sent several of my former pre-med students to their professor to get letters of rec; most of them complained that they barely knew the professor...So, I just told them that I would put in a good word with the prof. for them. But, really the professors pay more attention to the students than you probably think - I actually had a professor for a class with upwards of 600 students in it...he knew everything about me and I only talked to him once for 5 mins before class... Plus, the graduate students meet with the professors they teach for on a regular basis, if the professor doesnt know much about you, they will ask the grad student.
So, I suggest instead of asking the graduate student for the letter, I would ask the professor for the letter, make sure the professor knows which TA your taking class under and also tell the graduate student that you've asked the professor for a letter...see if you can get them to put in a good word for you.
In addition to what I said, I also agree with Shinydome. Ask profs first, but be sure to ask them if they could write a strong letter of recommendation for you. There would be nothing worse than getting a professor to write you a half-hearted LOR.
When I got my grad student letter, I had asked the professor first for a letter. Despite the fact that I had spent a lot of time with the professor in office hours, she said it was her policy to let the grad student write LORs to be cosigned by her. This worked out well for me since the course was a lab (so I saw the grad student for around 10 hours a week) and because I was on really good terms with the grad student.
The point is that professor letters are better but not getting one might not sink your application.