There are 3 different methods I use to solve the edges.

METHOD 1 - Cut across centers
Method 1 is manually pairing up the edges by cutting across the centers, moving those edges off to a 'safe' place and then undoing the damage you did to the centers. The only requirement is to make sure you have an unsolved pair of edges which you can 'sacrifice' when repairing the centers. The way I do this is to move the pieces I want to pair into the l and r slices and make sure I have an unsolved pair on either the Left or Right face.

In the example below I'm going to move the two yellow/blue edge pieces into the l and r layers then move the r layer up to pair with the other. I'll hold the cube so the pair is in the Front/Up position then rotate U' to move them to the Right face. Next you rotate R to put the yellow/blue pair in a safe place and bring the unmatched. At this point you rotate R until you bring up a pair of unmatched edges to the Up layer (I don't have any so I'm sacrificing the red/blue pair). Now it's time to reverse what we did earlier and repair the centers. Rotate the cube U and re-align the center pieces.

R'_D'_r_U'_R_U_r'

As you can see, the above method works as long as you have at least 3 messed up edge pairs, the 2 that contain the pair you want to match up and a third to sacrifice. Once you get down to just 2 unmatched edge pairs you'll have to go with Method 3 below. Also, in the 4th move where you rotate U' you could rotate U instead if you have an unmatched edge pair on the Left face you can sacrifice.

METHOD 2 - Diagonal edges
Method 2 is an algorithm that works pretty well. I probably use it more than I do Method 1. The process is to align a pair of edges in the corner of a face (I typically use the Front face) and then use the algorithm (Rr)_F_R'_F'_(Rr)'. This will line them up and place them in the Up/Front position. Note that you will destroy any matched pairs in the Front/Down position. To make this easier to see I am again using a semi-solved cube with just 2 pairs unsolved. I'll be pairing the red/yellow edges and consequently destroying the red/white pair at the bottom.

(Rr)_F_R'_F'_(Rr)'

Just like method 1, you can use the above method as long as you have at least 3 messed up edge pairs, the 2 that contain the pair you want to match up and a third to sacrifice. Once you get down to just 2 unmatched edge pairs you'll have to go with Method 3 below.

METHOD 3 - Opposing edges
Method 3 is an algorithm that works great, and the good news is that it does not destroy any previously matched pairs. To use this method I put 2 matching edge pieces into the Front face aligned in either the u or d layers. (The example shows both for reference only, it is not a requirement) Make sure you have the same color showing on the Front face with the other color on the Left/Right faces. You run the algorithm d_R_F'_U_R'_F_d' which puts the edges in the d layer into the Front/Right position and the edges in the u layer are paired into the Front/Left position. Again, I have started with the cube in a semi-solved state so you can see that none of the existing pairs are destroyed, only relocated.

d_R_F'_U_R'_F_d'

At this point all your edges should be matched up. You can proceed on to the Final Solve