If they don't light on the same side then the wiring is connected wrong. The front ones connect in the headlight shell and the rear ones connect under the seat, at the top of the fender. The fronts have 2 filament lamps, the dim one is for a marker light. When the ignition & headlights are on and low beam selected (assuming you have a headlight on/off switch, otherwise with ignition & low beam on) the markers should be on. Setting the turn signal switch turns the marker on the selected side off and the signal (bright filament) on through the flasher. The rear signal also goes on.
So make sure the wiring for all the signals and markers is correct so that the right marker goes out and the right signals go on when you signal right, same for left. Then you can go after the "relay".
As stated, normal auto flashers work fine except for the different shape. If you can find a 3-terminal flasher with power, ground, and load terminals that will work best; there's a green wire hanging loose beside the Honda 2-terminal flasher's mount that is perfect for the ground connection.