Fucoidans—the bioactive and structurally diverse polysaccharides from brown algae—have potential applications in the health industry, but development has been hampered by difficulties in controlling the structure, purity, and consistency of the product. Importantly, the extraction technique affects the nature of the product by preferentially extracting differing fucoidan compounds, whilst simultaneously altering their structure. Extractions are typically performed using hot, dilute acid, but the kinetics of extraction are poorly understood, especially with regard to trade-offs between yield, purity and structural integrity. Here, the extraction kinetics of fucoidan from the brown alga Ecklonia radiata under classical conditions (hydrochloric acid at 60 °C, pH 2) have been investigated, using both convective and microwave heating in open vessels for up to 3 h. The yield, purity and key structural characteristics of the fucoidan extracts were assessed. Only modest gains in yield were achieved beyond 6 min of extraction. Extended extraction was accompanied by undesirable declines in fucose and sulphate content, a substantial increase in the co-extraction of the contaminant laminarin, and significant reductions in the molecular weight of fucoidans. The results demonstrate that maximising fucoidan yield often comes at the cost of purity and structural integrity. It is important to tailor an extraction technique to best suit the desired applications of the final products. Open-vessel microwaves offer a highly efficient and controllable alternative to convective heating and merit further investigation, but techniques that better disrupt the cell wall complex whilst preserving fucoidan structure must also be pursued.