4 flashing led's

You're going to need more then relays diodes and resistors to make a control system like this.

First, why a circle?  Unless you have a very small vehicle no one will ever know that the pattern is a circle.  It may be more beneficial to just do alternating flashes (front left, rear right as one set and front right and rear left as the other).  This will give the illusion of a chasing pattern, but it will go in circles both ways around the vehicle.

Something like this would be very easy with a programmable processor and would require very little associated circuitry and draw very little current.

You may also want to look in to 555 timers and possibly some astable monovibrators.

I like the idea of alternating corners, it does make more sense, I just hadn't thought of it that way.

What kind of work are you talking about with that "programmable processor?" Also is there anyway I could get a diagram or anything posted?

There is a wide variety of programmable microcontrollers available.  A project this size is a great project to learn on - my first ever microcontroller project was actually an alarm LED controller.  It only had three lights though, but you could cycle through several different patterns with a pushbutton.  You can get a capable processor for about $2 (Atmel Tiny12) and spend about $3 making a parallel port programmer.  You can download the programming software (AVR Studio 4)and downloading software (Pony Prog) for free.  Then, spend a few hours learning program structure and the basic instructions and you're done.  (OK, I will admit, I over simplified it a bit, but anyone with a bit of determination would be able to get through it).  A great resource for beginners is www.AVRfreaks.net.

The circuit would be pretty basic - a 5vdc regulator, the microcontroller, a few caps, and the LEDs.  You should be able to drive the LEDs directly off the output pins of the controller.  Or, if you want to take a conservative approach you can add a transistor and 2 resistors for each LED.