A machine component comprising cooling channels through the interior of the machine component, the machine component being a component within a combustion turbine engine comprising one out of the following list of components: a turbine rotor blade; a turbine stator blade; a stationary shroud; a combustor liner; and a combustor transition piece, the cooling channels comprising: a first set of cooling channels, the first set of cooling channels including a plurality of parallel channels that reside in a first plane; a second set of cooling channels, the second set of cooling channels including a plurality of parallel channels that reside in a second plane; wherein: along a longitudinal axis, the cooling channels of the first and second set of cooling channels (24,26) comprise an alternating diverging-converging configuration, the alternating diverging-converging configuration creating a series of broader chamber sections connected by a series of narrower throat sections; the first set of cooling channels and the second set of cooling channels are configured such that, when viewed from the side, a crisscrossing pattern with a plurality of intersections is formed; the first plane resides in spaced relation to the second plane, with the first plane being offset from the second plane such that a plurality of the chamber sections of the first set of cooling channels connect to a plurality of the chamber sections of the second set of cooling channels; the chamber sections comprise channels of greater average cross-sectional area than the throat sections; and the machine component being characterized in that: the throat sections and the chamber sections consist of a smooth, curved contour; the throat sections comprise a converging section that narrows to a minimum throat area followed by a diverging section that broadens from the minimum throat area; and the chamber sections comprise a diverging section that broadens to a maximum chamber area followed by a converging section that narrows from the maximum chamber area. a first set of cooling channels, the first set of cooling channels including a plurality of parallel channels that reside in a first plane; a second set of cooling channels, the second set of cooling channels including a plurality of parallel channels that reside in a second plane; wherein: along a longitudinal axis, the cooling channels of the first and second set of cooling channels (24,26) comprise an alternating diverging-converging configuration, the alternating diverging-converging configuration creating a series of broader chamber sections connected by a series of narrower throat sections; the first set of cooling channels and the second set of cooling channels are configured such that, when viewed from the side, a crisscrossing pattern with a plurality of intersections is formed; the first plane resides in spaced relation to the second plane, with the first plane being offset from the second plane such that a plurality of the chamber sections of the first set of cooling channels connect to a plurality of the chamber sections of the second set of cooling channels; the chamber sections comprise channels of greater average cross-sectional area than the throat sections; and the machine component being characterized in that: the throat sections and the chamber sections consist of a smooth, curved contour; the throat sections comprise a converging section that narrows to a minimum throat area followed by a diverging section that broadens from the minimum throat area; and the chamber sections comprise a diverging section that broadens to a maximum chamber area followed by a converging section that narrows from the maximum chamber area. along a longitudinal axis, the cooling channels of the first and second set of cooling channels (24,26) comprise an alternating diverging-converging configuration, the alternating diverging-converging configuration creating a series of broader chamber sections connected by a series of narrower throat sections; the first set of cooling channels and the second set of cooling channels are configured such that, when viewed from the side, a crisscrossing pattern with a plurality of intersections is formed; the first plane resides in spaced relation to the second plane, with the first plane being offset from the second plane such that a plurality of the chamber sections of the first set of cooling channels connect to a plurality of the chamber sections of the second set of cooling channels; the chamber sections comprise channels of greater average cross-sectional area than the throat sections; and the machine component being characterized in that: the throat sections and the chamber sections consist of a smooth, curved contour; the throat sections comprise a converging section that narrows to a minimum throat area followed by a diverging section that broadens from the minimum throat area; and the chamber sections comprise a diverging section that broadens to a maximum chamber area followed by a converging section that narrows from the maximum chamber area. the throat sections and the chamber sections consist of a smooth, curved contour; the throat sections comprise a converging section that narrows to a minimum throat area followed by a diverging section that broadens from the minimum throat area; and the chamber sections comprise a diverging section that broadens to a maximum chamber area followed by a converging section that narrows from the maximum chamber area.