The best estimates at the moment are that the Universe is "flat". By that we mean that a pair of parallel laser beams will travel as far as you like and they will always remain the same distance apart and parallel.
Ironically, the best evidence for this comes from detailed measurements of the cosmic microwave background by the WMAP and now Planck satellites.
The problem you are having is you (I think) are imagining that the big bang happened at a point in space 13.7 billion years ago, and that radiation has been travelling outwards from that point. This is incorrect. Every point in space that we see now was, about 13.7 billion years ago, part of the big bang. As the space in the Universe expanded, its contents cooled and due to the trapping of free electrons by positively charged protons to form hydrogen atoms, the light that was emitted by the hot gas everywhere in the Universe at a time about 400,000 years after the big-bang was able to travel unabsorbed and unscattered. This radiation has travelled (at the speed of light) in all directions.
Now, when we observed the microwave background, we are seeing light that has been travelling in a straight line for just short of 13.7 billion years. That light was in the near infrared and visible part of the spectrum when it was emitted, but because of the expansion of space, the wavelength has also been stretched by a factor of 1100 to the microwave/short-wave radio part of the spectrum.