Bob Burton Pyraminx Method
This is the method I invented to solve the Pyraminx. It is extremely basic and doesn't require any algorithms. If you can solve the first layer of a 3x3x3 Rubik's Cube, you can solve a Pyraminx using this method without learning anything new. Even though this method is so simple, it is still good enough to achieve times of around 20 seconds on average and only about 30 seconds in the worst case scenario.
The Method

Step 1  Solving the TipsThe first step is to solve the trivial tips. Each of the four tips can be solved by just rotating them to match their adjacent pieces. 

Step 2  Solving the VerticesTo solve the vertices, just look at any tip. Since there are four colors on the Pyraminx and only three colors on each tip, the color missing from that tip will be the opposite face of that tip. Therefore, orient the other three vertices so that they are opposite the tip you chose. 

Step 3  Solving the First Layer EdgesTo solve the first layer edges, just insert them using three moves. This is similar to how you would insert a corner into the first layer of a 3x3x3 Rubik's Cube. The "algorithms" for this step are listed below. 

Step 4  Solving the Other LayersIf after Step 3 the puzzle is not solved, just rotate so that a different color is on the top face. Orient the final vertex of that color if it is not oriented already and solve the edges. One edge will already be solved from the layer you just completed in the previous step. If after solving that color layer the puzzle is not solved, choose yet a different color and repeat. You may have to go through all four colors to do this, but eventually the puzzle will be solved. Just be careful to only insert edges using the "algorithms" outlined below. Also make sure that you choose a different color each time you solve a new layer. 
"Algorithms"
I'd hardly call these algorithms, but here are the sequences needed to insert an edge into the first layer. For all algorithms on this page, I will be using Extended Pyraminx Notation.
Diagram  "Algorithm"  Comments 
R' D R  Perform this with red as the front face.  
L D' L'  Perform this with blue as the front face.  
L D L' D' R' D R  Perform this with red as the front face. It is just taking the corner out and inserting it as usual. 
Expansion
This method can be easily expanded to an intermediate method in which after the first layer is solved, you orient and permute the remaining three edges using an algorithm. There are only 5 cases and four of them can be solved using 3x3x3 algorithms.
Diagram  "Algorithm"  Comments 
R' L R L' (z60°) R L' R' L  Perform this with red as the front face (you should face the two flipped edges). The rotation is just a 60 degree rotation of the front face.  
R U R' U R U R'  This is just a Sune except the U2 is replaced with a U.  
R U' R' U' R U' R'  This is the same algorithm as the Sune except U is replaced with U'.  
L R U R' U' L'  Perform this with red as the front face. This is just the easy T orientation, also known as "Edges Bar."  
R' L' U' L U R  Perform this with blue as the front face. This is just the mirror of the easy T orientation. 