This is in no way meant to minimize the importance of game mechanics, their day will come soon, but I find myself drawn in by a theme before the mechanics, so that's why theme gets the first post.
Let's start at the beginning, shall we? Games like Candyland, Hungry Hungry Hippos, Chutes and Ladders, Hi-Ho-Cherryo, Mousetrap, etc., all have fun themes for young kids. They box covers are vibrant and engaging, and the game continues the motif. I doubt many of us would call those games with great themes, but they do engage the target audience.
Let's look at my top 20 games. Games with high theme are Cosmic, Battlestar Galactica, Eminent Domain, Star Trek Fleet Captains, Shadows over Camelot, Rattus, Forbidden Island, Pandemic, Catacombs, RTtA, Space Hulk, Small World, and 1960. That's 13/20. All the other games do have a theme, but they aren't as crucial to the game as the theme in the prior games.
Before I go any further, let's look at what theme does for a game. A great theme draws people in. You don't often hear someone say "come over and play that great roll and move game!" You hear "let's play that new space game, or that new civilization building game." The best of themes tell a story. They lead to stories you can tell days, months, even years later. You rarely hear a story about say Agricola that goes, "yeah, it was this awesome game, I won by 2 points because I got 1 more sheep than the other guy." You want grand tales to tell, like that one time in Shadows Over Camelot where everyone thought the traitor was going to win, and the knights pulled it out right at the end.
Now, theme caries a vast risk. If a game is built around a theme, several things can happen. First, the theme can fall flat for a group. I tried played the Game of Thrones game with my group last year, and it just didn't work for a couple reasons, but the theme was part of it. Being able to play the game at a high level was helped if one knew the source material. Second, if there are no mechanics to make the game interesting, you end up wanting a real game, regardless of the theme. For proof, look at the countless movie and tv tie-in games that exist with no real game behind them. Yes, there are some that shine, but the vast number of them end up on Goodwill shelves.
Theme isn't everything in a board game. If it were, the whole abstract genre wouldn't exist. Theme isn't even the most important thing in all games, but it is a nice bonus. Ideally, I like my games to have a mix of theme and mechanics. Basically, let there be a theme, let it not turn me off the game, and then have a solid game structure.