Kristine, I don't know what your electronics aptitude is, but mine was nearly as good as a rock or maybe a rotting log. I did some reading and some testing and was able to build my tail light project in a short amount of time. The pages below helped me the most.
I suggest using a 555 timer circuit that flashes the LEDs, then modify it to flash four and four with a couple transistors. You basically are flashing two LED's, one red and one white. I think you could then use a couple transistors in place of the single LED's to switch higher current and drive several LED's. The 555 will give you good control over the flash rates. If you used a variable resistor in the right place then you could "tune" the flash rate rather than guessing or doing a bunch of formulas to calculate it.
This page here has a 40 LED flasher all the way at the bottom that uses two transistors to handle the extra current for all those LED's. In your case you would just use 2 legs of 2 LED's on each transistor with a 270 Ohm resistor on each leg. It uses a single LED as a diode and that single flashes along with the others, but I don't see why you couldn't just keep that LED from being seen and let it do its' thing on the back of the sign. Just because its' there doesn't mean it has to be seen. I think this is probably the best circuit for your needs.
http://ourworld.compuserve.com/homepages/Bill_Bowden/555.htmTake a look at this site and look at the 555 timer circuit near the bottom of the page. It flashes two LED's alternately.
http://wild-bohemian.com/electronics/flasher.htmlUsing the LED Wizard on this website says to use a single 56 Ohm resistor to drive 4 LED's in series. You just wire your LED's in series when they are placed in the circle. I'm no expert on the 555, but it may even be able to drive all four LED's if you replace the 470 Ohm with the 56 Ohm. You may want to do some research on that, but if it works then you won't need the two transistors. I can't say anything about longevity. I have idea how to calculate that. But, you don't have much to loose by building the circuit and letting a battery run it until it dies and just see how long it runs.
I used this circuit, basically, to flash my brake LED's in my motorcycle tail light project, I'm just using one LED. If you want to look at my pictures you will see how I have placed a transistor where the single LED would have been and use that transistor to drive all 24 LED's, since the 555 wouldn't be able to drive them all by itself. Go to my Flickr page to see the photo's of the board,
http://www.flickr.com/photos/bluebeemerboy/sets/72157610821031491/You should be able to build something like this for less than $10, even if you buy all the components new from Radio S*ack or wherever. I salvaged every component in my project from electronics that were being discarded where I work. The only thing I bought was the circuit boards and wire.
If you need more help do a Google search on "555 LED flasher circuits" and you'll find no shortage of hits. I'll help you as much as my limited knowledge of electronics can.
I'm going to upload a picture of this 555 circuit showing the 8 LED's to the same set as the Flickr link above. This is just how I envision the circuit without transistors. You will need to confirm that the 555 can drive that much current, I just can't remember off the top of my head.
Other pages with several 555 circuits are
http://www.doctronics.co.uk/555.htm and
http://www.kpsec.freeuk.com/555timer.htmDon
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