You make no mention of the ultimate use these are serving. If they are under stress, like in the form of making a shelf, then glue may not hold well. Brads will likely spit the dowels unless you drill a hole for them first.

If it is indeed just an "art project" then yes, glue will hold. Use a good wood glue like Titebond. Set it all up on a flat table with a piece of wax paper under it, then glue it all down. Then put another layer of wax paper over it and then carefully lay some books on top of it to hold the dowels up against the board while drying.

Some things I have done sort of like this, I also used a few dabs of hot glue to hold them in place while the regular glue dries. Just put the glue on and leave a gap for the hot glue. Put a few dabs of hot glue in the spaces you left and press it down to your board. Hot glue sets in 30 seconds or less. Hot glue guns are cheap and you may already even have one. Hot glue may even do it ALL for you and won't require clamping it while drying.