A lot of the newer games are having more clever ways of determining who goes first. In my new game Gem Mining, I went with something cheeky and had the player who has been farthest west go first. The issue with this though is that he will always go first until someone decides to take a trip. For some games, this isn't a huge deal as the first player won't be given an advantage. The other thing this does is slow the start of the game. If you have to spend 5 minutes comparing to figure out who has the pointiest ears(Small World) then rolling the die would have been better. After that first game though, you know who goes first. However, he gets to start whenever he is there unless we use our trusty die. Old reliable d6.
For those games where the starting player has an advantage, it might be better to change it up more. There are some where it's the player who won last or the player to his left. "The player to his left" rule hopefully allows others to take advantage of the guy who won the last game. Nobody likes the guy who just won.
The topic has spawned new games even. There is Start Player from Z-Man games and a new kickstarter that uses dice called Start Player Express. These seem a little overboard but they do add a little flavor and something different. Also, if you have a group that makes arguments over the starting player, they could be of use.
If you are uncertain about what to do and don't want to roll dice, you can always do the tried and true method I use with my friends. Just ask, "Who's going first?" The response is either I am or they name someone else.
Remember, you're there to have fun. If you're having fun taking time to determine the start player, do it!
Play games. Always.