Blood is red

The primary cell in our veins is RBC (Red Blood Cells). Red Blood Cells are comprised primarily of a protein chemical called hemoglobin which contains the element iron (Fe). Iron is great at attaching to Oxygen, and when it does so it turns red.
When you get a nose bleed or have blood drawn it is a darker richer red because it is carrying more oxygen.

So what are the RBCs flowing around in? Plasma. Made of 90% water, making it mostly clear. You can have this drawn at a blood bank as well, and agree that it is mostly clear. A little opaque and gross looking, but not red (or blue for that matter).

So the argument for blood being blue comes primarily from observing ones veins. This means looking through your skin and the vein walls to determine the color. Therefor one must assume that both skin and the vein walls do not act like a colored transparency and change our perception...

Besides, I remember when the magic school bus took a field trip inside of one of the students who had a cold. I believe the river was red.

Sources:
The Franklin Institute.