Process (or workflow) execution on blockchain suffers from limited scalability; specifically, costs in the form of transactions fees are a major limitation for employing traditional public blockchain platforms in practice. Research, so far, has mainly focused on exploring first (Bitcoin) and second-generation (e.g., Ethereum) blockchains for business process enactment. However, since then, novel blockchain systems have been introduced - aimed at tackling many of the problems of previous-generation blockchains. We study such a system, Algorand, from a process execution perspective. Algorand promises low transaction fees and fast finality. However, Algorand's cost structure differs greatly from previous generation blockchains, rendering earlier cost models for blockchain-based process execution non-applicable. We discuss and contrast Algorand's novel cost structure with Ethereum's well-known cost model. To study the impact for process execution, we present a compiler for BPMN Choreographies, with an intermediary layer, which can support multi-platform output, and provide a translation to TEAL contracts, the smart contract language of Algorand. We compare the cost of executing processes on Algorand to previous work as well as traditional cloud computing. In short: they allow vast cost benefits. However, we note a multitude of future research challenges that remain in investigating and comparing such results.