Narrowband Internet of Things (NB-IoT) is a new narrowband radio technology in fifth-generation (5G) networks. In NB-IoT cellular networks, to provide low-power wide-area coverage, there are several different resource allocation units that can be allocated by a base station to NB-IoT devices for uplink transmissions. Traditional resource allocation algorithms without considering the multiple resource allocation units are not appropriate for NB-IoT networks, and we observe that only adopting the same resource unit for each device will result in the radio resource wastage. Therefore, this paper investigates the uplink resource allocation problem with the considerations of the new radio frame structure and multiple resource units for NB-IoT networks. The objective is to minimize the used radio resources while each device can transmit its data. We propose an algorithm to determine a suitable resource unit and allocate the radio resource for each device to solve the target problem. Compared with a baseline, the simulation results show the efficacy of the proposed algorithm and provide useful insights into the resource allocation design for NB-IoT systems.