What is a simple yet efficient way of converting 12VAC to 5-12VDC?
For converting mains power to DC, "Full wave rectifier + filtering caps" is the only realistic option. 60W / 12V = 5A, which is easily achievable with reasonable size components. Your LED driver can handle a wide input voltage range and regulates LED current, so an unregulated supply should be OK.

For best efficiency you should use high current Schottky diodes and low ESR filter capacitors. The only question is what ratings and values are required? Incorporating all relevant factors makes the calculations a bit tedious, so I simulated the following circuit in LTspice:-







simulate this circuit &ndash; Schematic created using CircuitLab

Peak AC input voltage to the rectifier was 16.6V (blue line). The diodes dropped 0.5V each so the peak DC output voltage was 15.6V (green line). Peak diode current was 24.6A (red line). Between peaks the capacitors held the DC voltage up, but it dropped as the load current discharged them. With a total of 9400uF (2 x 4700uF 25V) the minimum 'trough' voltage was 12.6V, corresponding to 63W minimum output power. 

The average mains input power was 81.1W (ignoring transformer core loss which wasn't simulated) and the average output power was 70.6W, for an AC-DC conversion efficiency of 87%.