Bouquet garni is basically a bouquet, not of flowers, but of a few herbs. Cooked long in a stew they add a wonderful bouquet in the aroma sense like wine-lovers use it.
Classically, the herbs are tied with cotton string or wrapped in cheesecloth (which allows you to use loose dried herbs) and then tied. One easy alternative may the storebought, variety, already assemble in a sort of teabag.
If you decide to make your own, the traditional herbs for boeuf bourguigon are a bay leaf, a sprig of fresh thyme, and parsley stems (which you reserved from chopping the leaves).
I'm not saying the classical methods are a conspiracy of classically trained chefs but, arguably, you can get the same flavors from the individual herbs simply tossed loose into the stew. Simply fish out the bay leaf and parsley stems before serving if you want. Same for the thyme sprig, or if you use dried thyme, who's going to criticize you leave it in. Call it "rustic style." Or tell your guests that it is a family tradition (going back... gosh, minutes) to leave the bay leaf in and it is considered good luck if you get in in your serving.
Whether you decide to go classic, storebought or rustic, Beouf Bourguignon is the perfect recipe to use it.
For suggestions on using other ingredients common to French cuisine, check out Classic French Ingredients .