Learning to capture text-table alignment is essential for tasks like text-to-SQL. A model needs to correctly recognize natural language references to columns and values and to ground them in the given database schema. In this paper, we present a novel weakly supervised Structure-Grounded pretraining framework (STRUG) for text-to-SQL that can effectively learn to capture text-table alignment based on a parallel text-table corpus. We identify a set of novel pretraining tasks: column grounding, value grounding and column-value mapping, and leverage them to pretrain a text-table encoder. Additionally, to evaluate different methods under more realistic text-table alignment settings, we create a new evaluation set Spider-Realistic based on Spider dev set with explicit mentions of column names removed, and adopt eight existing text-to-SQL datasets for cross-database evaluation. STRUG brings significant improvement over BERTLARGE in all settings. Compared with existing pretraining methods such as GRAPPA, STRUG achieves similar performance on Spider, and outperforms all baselines on more realistic sets. All the code and data used in this work will be open-sourced to facilitate future research.