Go out on a Sunday when there is little traffic and the city is pretty much yours. Cycling by yourself perfectly fine, you avoid the same areas as you'd avoid anyway. Sunday traffic, especially around lunch hour, means no one around. It depends on what you are looking for as non-touristic. I like going around different neighbourhoods and just seeing where I end up, tend to just follow a bike path to see where it dumps you (and they often do just end in the middle of nowhere).
http://lavidaenbici.com/buenosaires/ -- shows you some of the path and calles recommendadas
I used to go from Las Canitas along by the golf course, up figueroa alcorta, over the hwy pass at the Tiro Federal (Udaondo) down the path along Cantilo out to the Kids park/Naval area, into the Ciudad Universitaria out through the back and onto the Costanera, past Aeroparque, cross the street (which is the bit that's dodgy) and then back into the city via Sarmiento or Salguero depending on whether or not I wanted a slightly more extended loop. Then in via the ciclovias on Figueroa Alcorta or near the Planetarium, along the cyclepaths by the parks and the Rosedal, back into the golf course and home. It's a nice ride with only a couple of hairy bits when crossing traffic, but for the most part there are cyclepaths the whole way or very little traffic. I think even by Aeroparque now they have finished the cyclepath that goes under the tunnel back to the Planetarium, though it dumps you in the middle lane of Sarmiento and then you have to make sure there's no crazy drivers entering there at 100km off the hwy before you cross over.
From the Kids Park you can also continue through there and out the other side into Vicente Lopez and explore into the Zona Norte.
From that same area I also used to enjoy just going towards Chinatown into Nunez, up to Parque Saavedra, into Villa Urquiza, along one of the cycle paths there and then back via Belgrano -- La Pampa has a ciclovia now but on weekdays it is jam packed with trucks parked in it, and parts of it are still under construction, so I tend to just go alternate paths.
For racing enthusiasts there are always huge groups of cyclists around Villa Urquiza doing a loop that goes around Parque Sarmiento, down Larralde, out along the collector. I think a lot of them also go over the hwy pass and into provincia to make a larger loop. They are out there every day but on weekends it is hoardes. You have to be daring as you're racing with a lot of the buses and crazies down the collector, but I've done it on my crapola bike and it's a good circuit, though these guys with actual racing bikes of course whizz by me and make me feel like a loser
