I'm not going to spend a whole lot of time talking about my bicycling background; if you know me, you know that I cycle practically everywhere, and Tbilisi is not an exception. However, the reaction by Georgians and expats alike when they see me using a bike as transportation is usually something along the lines of "Wow, you're brave! It must be pretty dangerous out there, huh?"
This is a common reaction because drivers in Tbilisi are widely acknowledged to be crazy--Georgian machismo combined with lax traffic enforcement leads to dangerous speeding, aggressive tailing, games of Chicken, and lots of accidents. In six months in Georgia, I've personally witnessed two rear-end collisions, had the city bus I was riding in scrape the side of a taxi, and driven or walked past three multi-vehicle high-speed accidents. Not the biggest sample size, of course, but I'm pretty sure the statistics support me on this one.
Despite this, I actually feel pretty safe riding my bike around Tbilisi, for a few reasons.
The first reason is simply experience: I've ridden thousands of total miles on dense, high-traffic streets in Chicago, and while Chicago is a very bike-friendly city by American standards, it has its fair share of crazy drivers. Knowing how traffic flows, and how to position yourself in the road is a key part of safe city cycling (I suggest The Art of Cycling to anyone who wants to learn more about city riding; I haven't read it, but based on the review, it sounds comprehensive and agrees with pretty much everything I would teach a beginning city cyclist).
The second reason is novelty: There are plenty of cyclists in Tbilisi, but they tend to fall into two categories. One group is teenage boys who ride bikes in order to look cool. They generally ride hideous bicycles that are apparently designed to look like monster trucks, with lots of superfluous shock absorbers and needlessly massive (and heavy!) tubing arranged in the most aggressive-looking way possible. These guys basically ride around on the sidewalk, weaving through pedestrians and generally being idiots. The other group are recreational riders, who use their bikes for sports, like mountain biking, or touring through Georgia's gorgeous mountain roads.
However, there are precious few people who use bicycles as transportation, on city streets, on a daily basis. In an American city, this is generally a bad thing. When American motorists aren't used to looking for cyclists, they're more likely to hit them, because cyclists appear in different places than cars do. However, in Tbilisi I find that the rarity of bicycles on the road is actually a good thing. The drivers here are aggressive, but they're also attentive--all the chaos here forces you to be alert when you drive. Drivers here are more likely to see cyclists because they're watching the whole road, and when they do see you, the novelty really helps--drivers generally freak out a little bit when they see a bicycle, and will give you space in a way that they don't in Chicago where awareness of bicycles is higher.
The final reason is the lack of animosity. In most American cities, there is a certain group of drivers who actively resent the encroachment of cyclists onto "their" roads. These guys will deliberately make cyclists' lives miserable, simply for being on the road. They will cut in front of you, honk their horns to try and scare you, or spray you with windshield cleaning fluid (I'm not the kind of cyclist to key someone's car or bash their windows with a U-lock, but I was pretty close with the windshield wiper guy). I think that this attitude is only possible because American roads are so pleasant to drive on and American drivers so law-abiding--in Tbilisi, the average driver has to contend with so many things getting in their way that anyone who flew into a frothing rage at the slightest infringement on "their" patch of pavement would get arrested instantly. Once again, chaos makes for safer biking--rather than an "intruder" into the automobile's rightful domain, drivers view cyclists as simply another obstacle to be avoided.
I wouldn't recommend street cycling in Tbilisi to everyone -- it is often chaotic, and it's challenging to do safely. But that's true about every city I've ever cycled in, so if you're an avid bicycle commuter already, Tbilisi's not so different.