Example methods and apparatus to measure fluid flow rates are disclosed. A disclosed example apparatus includes a circulator to selectively circulate a fluid in a flowline, a generator thermally coupled to the flowline at a first location and controllable to form a heat wave in the fluid, a sensor thermally coupled to the flowline at a second location to measure a first value representative of the heat wave, a phase detector to determine a second value representative of a wavelength of the heat wave based on the first value, a frequency adjuster to control the generator to form the heat wave in the fluid at a first frequency, the first frequency selected so that the second value is substantially equal to a distance between the first and second locations, and a flow rate determiner to determine a flow rate of the fluid based on the first frequency.