This is hard to do if you didn't include a shunt resistor or something. It's not so bad if you can measure a trace resistance and voltage drop, but big power pours shoot that down.
One way is to connect a DC power supply across the voltage rail on the board and tease the supply voltage up until it takes over from the regulator on the board, and note the supply current.
But if the supply is a buck switcher, it may decide not to shut down, but to load the over-voltaged rail and push power uphill. That messes things up. And burp-mode switchers make it even more interesting.
For a switcher, you can measure the output inductor resistance, then measure the running DC voltage drop. Assuming your DVM can accurately measure low resistances, and then measure small voltages in the presence of many volts of high-frequency AC and spikes and stuff.
My Fluke is measuring 920 uV across the inductor in a +12 to +1 volt switcher, which corresponds to 36 mA of FPGA core current. I measured that an other way and got 50 mA, which for a measurement like this ain't bad.