Not sure about too many birthday games but you could try:
-worksheets, (counting birthday items, wordfind)
-the birthday song, it's easy and catchy
-making a b-day card,   

Some games you can do with b-day flashcards are:
-memory game- display the flashcards and have the students remember them. Turn some over and have them try to remember the whole list.  Eventually you will have them all turned over and the students may or may not remember the whole list.  You can have teams compete against each other.   
-Alternative version, take a flashcard(s) away and they have to guess what card it is.  These even work well in my middle school classes for remembering vocab.
-not sure what your classroom is like or if running will work but you could hang flashcards in the classroom and call out one.  The students must race to slap the correct card.  Winner, gets a point for his/her team.
-alternative non-running version- involves a fly swatter and placing them on the board.  They have to hit the correct one fastest to get a point for their team. 

If you want to make it really fun, buy one of those colorful, plastic, squeaking hammers at homeplus.  You can bonk the losers on the head or have the winners bonk the losers on the head.  ( I miss hitting my kindergarten kids with the hammer, good times) 

one more stolen from Daveseslcafe.com
Birthday Bingo
(may need to simplify but could work) 
variation of bingo to practice ordinal numbers and to personalize the game for the students.
Pre-teach dates and ordinal numbers (get them to line up and rearrange them a couple of times!). In pairs, then to the class, students say their birthdays. Then have students come up to the board and write down their birthdays in number form (for example, 8/12/80). Students draw the grid of squares (3x3) and fill it with 9 birthdays of their choice. Then the teacher, or a student, reads out birthdays off the board in full (for example,'eighth of december nineteen-eighty'). The students cross off the birthdays they hear and first student with a correct row horizontally, diagonally or vertically wins.
You could also do this with times.
From Nic, Zhengzhou, China, courtesy of Dave's.
 
Cheers

Here's a matching game. 

If you're doing a birthday class here is a pin the tail on the donkey game.

A presentation with vocab and examples and a fill-in worksheet for the first 2 dialogs.

Because Mediafire is gone, I'm gonna upload my stuff here :]

Some of it's mine, some of it I got from Mediafire and fixed, or some of it is just a blatant steal from mediafire ;].

This is for the green book with the globe on it
Worksheet with dialoge and some games

thanks for the pin the tail on the donkey.  i did this last year and it was a total success.

Great version if you plan on teaching the traditional Birthday Song.

http://www.gfes.tpc.edu.tw/board/abc-song/birthday.swf

When is your birthday ppt...good for conversation.

well...here is a Hidden picture game for this Chapter.  Mainly Lesson 2 when they introduce the phrase "Is it a ____?  I got the template from www.eslgamesworld.c om