We design and assess novel random access (RA) procedures for use in 5G radio access networks in support of machine-type applications requiring Ultra-Reliable Low-Latency Communications (URLLC). In a simulation-based performance assessment the proposed RA procedures are compared against LTE Cat-M1 (or LTE-M), the current LTE-based access technology for such applications, considering a Factory of the Future scenario and a realistic density and mix of URLLC and non-URLLC devices. As a key outcome, the most promising Two-step RA with Feedback procedure is demonstrated to be able to satisfy the 3GPP-imposed 10 ms (50 ms) latency requirement of URLLC (non-URLLC) applications in 99.901% (99.997%) of the cases, with a theoretical minimum access latency of only 5 ms.