Get to Convict Hill and Mopac, carry your bike over the barrier and get into the Mopac turnaround, and take Mopac north.  Stick to the frontage roads, but you'll have to take the lane on Mopac proper as you go over the greenbelt and you're mingling with 65 mph traffic.  That part is only 0.6 miles long, and it's slightly downhill so you can easily do 40 mph, but it ruins what would otherwise be a reasonably acceptable route -- the frontage roads on Mopac aren't too bad.  After that, you just stay on Mopac's frontage roads until you hit Barton Springs.

That said, if you want to go mountain biking, I imagine you could just go down into the greenbelt before that point.  It would be slower, but no cars.

This would shave several miles off the routes that others have given you, depending on exactly where you want to go.  (For example, I live right near Dick Nichols and work at Barton Skyway and Mopac.  If I take that route to work, it's right at six miles total.  If I take the routes the others have mentioned, it's around 12-13 miles.)

The city is building a pedestrian bridge there -- they just started construction a few weeks ago -- but it's still at least a year away from completion.  I imagine once that's done, they'll put in bike lanes or something else along the entire route.  There's also a trail that's being built or planned, called the Violet Crown trail, that will pretty much go from Dick Nichols to Barton Springs, but I don't know what the timescale is for that or what it'll be like or what.

For the return trip, over the greenbelt is a lot less hairy, as you have a shoulder, but the frontage roads are a little more hairy than they are on the other side (but still nowhere near as bad as the 0.6 miles over the greenbelt.)

I should add that I am a pretty confident cyclist and I don't mind taking some dirt trails and busy roads if needed. I just want to avoid any suicidal roads. I am also thinking of taking my mountain bike some time and riding to the Greenbelt and riding singletrack part of the way. that might be more work than it's worth just to get to work but it could be fun.

Slightly longer but absolute minimum of traffic. Much of this ride you can go several blocks without encountering a moving car.  You have to carry bike across R tracks at one point:  http://www.mapmyride.com/routes/view/366589303

Don in Austin

I used to live there and biked the same thing.  This is the route that I used:

http://goo.gl/maps/GM8rE

About 8.5 miles, and Brodie is about the only busy street you're on.  Brodie has had good bike lanes for a couple of years, and is otherwise abnormally wide.  Taking Lamar directly instead of the Bouldin Creek neighborhood cuts off about one mile.

My route is a bit straighter.  Still pleasant.  http://goo.gl/maps/5ZlPq

Don in Austin

http://goo.gl/maps/GM8rE

About 8.5 miles, and Brodie is about the only busy street you're on.  Brodie has had good bike lanes for a couple of years, and is otherwise abnormally wide.  Taking Lamar directly instead of the Bouldin Creek neighborhood cuts off about one mile.